### The Jacobian conjecture and the extensions of polynomial embeddings.

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We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets ${U}_{i}$ which are isomorphic to closed smooth hypersurfaces in ${\u2102}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {\u2102}^{m}$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{\u2102}^{n}\to {\u2102}^{m}$ such that $X\subset {S}_{f}$. This gives (together with [3]) a full characterization of irreducible components of the set ${S}_{f}$ for generically-finite polynomial mappings $f:{\u2102}^{n}\to {\u2102}^{m}$.

We describe the set of points over which a dominant polynomial map $f=({f}_{1},...,{f}_{n}):{\u2102}^{n}\to {\u2102}^{n}$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod}_{i=1}^{n}deg{f}_{i}-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_{i}\right)$.

We study some properties of the affine plane. First we describe the set of fixed points of a polynomial automorphism of ℂ². Next we classify completely so-called identity sets for polynomial automorphisms of ℂ². Finally, we show that a sufficiently general Zariski open affine subset of the affine plane has a finite group of automorphisms.

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

Let X be a smooth algebraic hypersurface in ℂⁿ. There is a proper polynomial mapping F: ℂⁿ → ℂⁿ, such that the set of ramification values of F contains the hypersurface X.

Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f=(f\u2081,...,{f}_{m}):X\to {k}^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function...

We give a simple geometric proof of Mohan Kumar's result about complete intersections.

Assume that X,Y are integral noetherian affine schemes. Let f:X → Y be a dominant, generically finite morphism of finite type. We show that the set of points at which the morphism f is not finite is either empty or a hypersurface. An example is given to show that this is no longer true in the non-noetherian case.

Let $(X,{\omega}_{X})$ and $(Y,{\omega}_{Y})$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $\Phi *{\omega}_{Y}=c{\omega}_{X}$.

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